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\pi + \sqrt2 transcendental if it is known that \pi is transcendental

Source: Chile Finals 2008 L2 p6

October 5, 2022
algebratranscendental

Problem Statement

It is known that the number π\pi is transcendental, that is, it is not a root of any polynomial with integer coefficients. Using this fact, prove that the same is true for the number π+2\pi + \sqrt2.